Title: The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems

Abstract:

Diffusion models and flow matching techniques are becoming standard tools for addressing partial differential equation (PDE) inverse problems. In these applications, practitioners typically enforce governing physical laws as a strict constraint through projection or guidance mechanisms, subsequently interpreting the generated samples as a Bayesian posterior with properly calibrated uncertainty. However, we demonstrate that this prevalent methodology yields samples from an incorrect distribution.

Conditioning a generative prior on a hard PDE constraint effectively conditions on a manifold of measure zero. This operation is inherently ambiguous, a phenomenon known as the Borel–Kolmogorov paradox. The physically accurate resolution involves taking the small-residual-noise limit, which introduces a co-area (Fixman) Jacobian factor of $[det(JJ^{\top})]^{-1/2}$. Crucially, standard projection- and guidance-based approaches inadvertently omit this factor.

We quantify this bias, demonstrating that it intensifies as the heterogeneity of constraint sensitivity increases. Through validation on controlled problems against an independent and identically distributed (i.i.d.) ground-truth arbiter, we highlight the severity of this omission. It is not merely a minor detail: failing to include this factor inflates posterior error to $20\times$ the sampling-noise floor. Even minimal-displacement projection methods, such as those used in PCFM, exhibit a bias of $9\times$ the floor, while naive scalar reweighting fails to correct the issue.

To address this, we introduce CoCoS, a measure-aware constrained sampler designed to target the correct co-area posterior. Our experiments show that CoCoS aligns with the gold-standard posterior within the limits of sampling noise. These findings underscore that "satisfying the physics" is distinct from "sampling the posterior," offering a principled correction for uncertainty-aware scientific inference.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC